Anand CV Mallaya

Chaos theory and the evolution of sexes

In evolution, life, math, science on January 24, 2012 at 9:29 pm

Chaos theory is considered by many modern mathematicians and scientists as one of the  four pillars of modern science. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the butterfly effect. Chaos theory has  applications in multiple scientific disciplines including:  geology, mathematics, microbiology, biology, computer science, economics, engineering, finance, meteorology, philosophy, physics, politics, population dynamics,psychology, and robotics. This article is an examination of the significance of chaos theory  in the evolution of sexes in sexually reproducing species including humans.

Chaos Theory

Chaos theory in simple terms states that dynamical system which are deterministic (having finite numeber of states and depends only on the initial condition of the system) are unpredictable by nature. In real world this has big consequences because many systems in nature are extremely sensitive to initial condition (such as weather,stock markets,chemical reactions etc.) and which makes it very difficult to predict and control.

Bifurcation and strange attractors

One interesting property came from the study of chaotic systems is that there is a phenomenon called strange attractors. Some systems tends to be chaotic with most of input space(called phase space) except certain region of the phase space. The diagram below shows the two attractors of the system represented by the logistic equation x = 4 x (1 – x) [image courtesy] In many of the systems this process tends to be centered around two attractors. And this is called bifurcation process. This phenomenon is of interest and significance in this article. [image courtesy]

Evolution of sexes

Evolution of sexual reproduction and sexes are of great interest among modern biologists and is one of the mysterious area in biological evolution. There are many competing theories trying to explain this process. Here the attempt is to give a mathematical dimension to the search. As we now, through evolution, living organisms takes all possible forms and tactics to survive and propagate. In most of the species, it is striking that the number of sexes are limited to two(male and female). This is either by chance or limited by certain constraints exerted by nature(like mitochondrial efficiency in the nucleus). But the number two gives rise to the question of its relationship with mathematics. Here are some patterns that connect the chaos theory with evolution

  1.  The biological system(morphology) of any organism is dynamical system.
  2.  Evolution process is mostly in the change in DNA.
  3.  Sensitivity of the morphogenesis of organisms to initial conditions(the DNA).
  4.  Sensitivity of the sexual dimorphism of organisms to initial conditions(X,Y chromosome region of the DNA)
  5.  Large number of iterations over time to support bifurcation(the evolutionary timeline is of the order of billions of years).
  6.  The evolutionary process tries to takes all forms(phase space of the life form) based on the change in DNA(initial condition)
  7.  The number of sexes tends to attract to two(strange attractors,bifurcation?).

The figure below show the hypothetical bifurcation of the evolution of x and y chromosomes. Start from a common ancestor of chromosome pair x’x’ the ancestors of x and y sex chromosomes. Assuming there were no sexual dimorphism at this phase and the evolutionary change can be in any direction. But the number of sexes converges to 2 rather than n types able to reproduce with each other leading to xx(female) and xy(male).

bifurcation of sex

hypothetical bifurcation of the evolution of sexes

Put your valuable comments below or discuss it in Quora The chaos theory of evolution:  An interesting article published in in New Scientist discusses similarity of evolution of life and non-linear systems, fractal nature of life and chaos. My own conversation in ideas :

Interesting Reads 

Some articles which are good read for good background of concepts in Evolutionary Biology, related to Sex:

  1. Evolutionary origins of Cooperators and Defectors
  2. Cooperation and the evolution of anisogamy
  3. A chapter from the book The Score by Faye Flam
  4. Sexes redefined @
  5. The chaos theory of evolution

Interesting videos

During further research, I got these resources which are enlightening some aspects of the truth. Reason yourself.

Evolution of Sex

Origin of Sexes

Mathematical Model of Evolution

Chaos theory

Other articles linking to this page

  1. While there are good arguments for the value of sexual reproduction, I don’t know if there a large benefit for a second bifurcation to four nodes or four genders. I suspect that a four gender form of life would add a level of fragility into life such that reproduction would cease at a much higher minimum population density, thus offsetting any benefit of the increased DNA mixing.

    • Dear Sir,

      Thank you for taking time to post your thoughts here.

      A four-sex system can be predicted, if we can prove that the process is in fact chaotic (I am in active research on that). Whether such a four gender system is stable in terms of fitness will determine their presence in the population. If the hypothesis can be proved, we can predict when will it emerge as well. Since two-sex system has many advantages over asexual reproduction, it gave raise to complex organisms leading to humans. We may be still in the process of evolution and it may well be very early to develop next bifurcation.

      On the other hand, my research lead to one single point on why we have two sexes rather than multiple. The answer is closely associated with why most sexually reproducing organisms have two pair of chromosomes rather than many(diploid genome compared to haploid or higher order Ploidy).

      This is what I am focusing at the moment. What I suspect is that, its because, its sufficient to provide stability and advantages of sexual reproduction) and that originated when early single celled organisms exchanged genes through methods like Horizontal Gene Transfer etc.

      I would love to hear educated thoughts on it.

      Anand Mallaya

  2. One of the concepts that seem to apply here is that “Life is more vibrant at the edge (of chaos).” As such, two node sexual reproduction is more vibrant or resiliant than asexual reproduction.

    It occurred to me that there might be some variant of sexual reproduction that goes beyond bisexual methods. For the initial conversion from asexual to bisexual reproduction, early on, there had to be some mechanism of gene incorporation from another ogranism. That process eventually became common practice over time. We still have organisms that conjugate once and then divide dozens of times before another conjugation is required to restart the process. I believe that is the case for some of the obligate protozoal intracellular parasites. I suspect that a similar process is at work even in human cells wherein they are able to reproduce to some Hayflick number of 60 divisions before their teleomere depletion causes them to cease replicating.

    There is still aspects of the early asexual reproduction with gene sharing in bacteria where they incorporate pieces of DNA from other bacteria without it being a sexual act. That is how some bacteria acquire drug resistance just by being grown in the presence of resistant bacteria of a different strain.

    That said, there may be forms of real but infrequent four node sexual reproduction of some microscopic organisms. From my previous work with protozoa, I know that generating the conditions for two node reproduction is not easy. My guess is that in the wild, there are occasions of four node reproduction that we are not yet able to produce in the lab. I would guess that even higher node reproductive events also occur.

    It is hard enough to catch many unicellular and multicellular organisms in the act of traditional conjugation. We may have seen four node reproduction, either as a sequence of closely spaced two two-node conjugations or as a simultaneous four node conjugation, but not recognized it simply because we are not looking for it, or we are constrained conceptually by our two node mental framework.
    Have fun,

    • Hi Edward,
      I somehow missed this comment for a couple of years, only to discover it today by chance!

      Its astonishing to see that you really think that four node conjugation can occur in the wild, which we have never tried to look for!
      In my world view, it can happen. Where should I look for it? Can you give me some advise on how can I design some experiments to hopefully observe and record such interaction (I am not trained in the life sciences apart from high school education). I feel it very fascinating nonetheless – watching ‘The greatest show on earth’


  3. Hello Anand,
    I think four node reproduction may not look like what we would expect, from our 2 node perspective. It is like in the book Flatland, where a two dimensional animal can not conceive of a three dimension world.

    As such, we already exist in a mixed one and two node reproduction model. Our chromosomes are two stranded, and they come from two different parents. The mitochondrial DNA however, comes only from the mother, so mitochondria are a type of one node reproduction.

    As far as direction to look, I would start with smaller organisms first, such as single cell organisms since they are often found in higher densities making bumping into other organisms much more likely.

    If we consider multiple node cooperation as a form of multiple node reproduction, explore sperm to egg behavior. When last I studied that area, it was considered that the egg must be exposed to multiple sperm before it became receptive to one sperm to enter. This may be why a male is considered infertile if the sperm count falls below a certain number.

    There are some single cell organisms (bacteria) that share DNA once, and are capable of reproducing many times before they need another exchange of DNA. This too is a variation somewhere between single and double node reproduction.

    Leishmania Donovani, a tropical disease organism, spread by the sand fly displays a clustering behavior in its extracellular flagellate stage. At the time I was studying that organism the stage of their life cycle at which they shared DNA was not known. It occurred to me that highly motile cells need to attach to another cell to share DNA with that cell, and this might be one form of this process. The fact that it resembled the human perm-egg facilitating behavior made me wonder if this was a group sex activity.

    Another point to consider is that a three or four node conjugation requirement for reproduction does not mean that it has to happen at the same time. A sequence of an organism paring with one partner, followed by the same with another can include the passage of time between the conjugations.

    The cooperation of multiple nodes contributing to reproduction could also be opportunistic as opposed to mandatory. Again, back to bacteria, the incorporating available free strands of DNA found in the growth medium would be one form of this opportunistic multi node reproduction.

    As a conclusion, our concept of two node reproduction is influencing our concept of what multi-node reproduction must look like. We should not be constrained by the paradigm we have come to know and love.
    Have fun,

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